

SMARANDACHE SEQUENCE OF HAPPY NUMBERSShyam Sunder GuptaChief Engineer (C), S.E. Railway, Garden Reach Kolkata  700043, India Email: guptass@rediffmail.com
Abstract: In this article, we present the results of investigation of Smarandache Concatenate Sequence formed from the sequence of Happy Numbers and report some primes and other results found from the sequence Key words: Happy numbers, Consecutive happy numbers, Hsequence, Smarandache Hsequence, Reversed Smarandache Hsequence, Prime, Happy prime, Reversed Smarandache Happy Prime, Smarandache Happy Prime
3.1 Observations on Smarandache Hsequence: We have investigated Smarandache Hsequence for the following two problems.
In search of answer to these problems, we find that
It may be noted that SH(10000) consists of 48396 digits. Based on the investigations we state the following: Conjecture: About oneseventh of numbers in the Smarandache Hsequence belong to the initial Hsequence. In this connection, it is interesting to note that about oneseventh of all numbers are happy numbers [1].
3.2 Consecutive SH Numbers: It is known that smallest pair of consecutive happy numbers is 31, 32. The smallest triple is 1880, 1881, 1882. The smallest example of four and 5 consecutive happy numbers are 7839, 7840, 7841, 7842 and 44488, 44489, 44490, 44491, 44492 respectively. Example of 7 consecutive happy numbers is also known [3]. The question arises as to how many consecutive terms of Smarandache Hsequence are happy numbers. Let us define consecutive SH numbers as the consecutive terms of Smarandache Hsequence which are happy numbers. During investigation of first 10000 terms of Smarandache Hsequence, we found the following smallest values of consecutive SH numbers: Smallest pair: SH(30) , SH(31) Smallest triple: SH(76), SH(77), SH(78) Smallest example of four and five consecutive SH numbers are SH(153), SH(154), SH(155), SH(156) and SH(3821), SH(3822), SH(3823), SH(3824), SH(3825) respectively. Open Problem: Can you find the examples of six and seven consecutive SH numbers? How many consecutive SH numbers can you have?
4.0 Reversed Smarandache HSequence: It is defined as the sequence formed from the concatenation of happy numbers (Hsequence) written backward i.e. in reverse order. So, Reversed Smarandache Hsequence is 1, 71, 1071, 131071, 19131071, 2319131071, 282319131071, ... . Let us denote the n^{th} term of the Reversed Smarandache Hsequence by RSH(n). So, RSH(1)=1 RSH(2)=71 RSH(3)=1071 RSH(4)=131071 and so on.
4.1 Observations on Reversed Smarandache Hsequence: Since the digits in each term of Reversed Smarandache Hsequence are same as in Smarandache Hsequence, hence the observations regarding problem (ii) including conjecture mentioned in para 3.1 above remains valid in the present case also. So, only observations regarding problem (i) mentioned in para 3.1 above are given below: As against only 3 primes in Smarandache Hsequence, we found 8 primes in first 1000 terms of Reversed Smarandache Hsequence. These primes are: RSH(2) = 71 RSH(4) = 131071 RSH(5) = 19131071 RSH(6) = 2319131071 RSH(10) = 443231282319131071 Other three primes are RSH(31), RSH(255) and RSH(368) which consists of 72, 857 and 1309 digits respectively. Smarandache Curios: It is interesting to note that there are three consecutive terms in Reversed Smarandache Hsequence, which are primes, namely RSH(4), RSH(5) and RSH(6), which is rare in any Smarandache sequence. We also note that RSH(31) is prime as well as happy number , so, this can be termed as Reversed Smarandache Happy Prime. No other happy prime is noted in Reversed Smarandache Hsequence and Smarandache Hsequence.
Open Problem: Can you find more primes in Reversed Smarandache Hsequence and are there infinitely many such primes? [1]. Guy, R.K., "Unsolved Problems in Number Theory", E34, Springer Verlag, 2nd ed. 1994, New York. [2]. Marimutha, H., " Smarandache Concatenate Type sequences", Bull. Pure Appl. Sci. 16E, 225226, 1997. [3]. Sloane, N.J.A., Sequence A007770 and A055629 in " The on line version of the Encyclopedia of Integer Sequences". http://www.research.att.com/~njas/sequences/. [4]. Weisstein, Eric W, "Happy Number", "Consecutive Number Sequences" and "Smarandache Sequences", CRC Concise Encyclopedia of Mathematics, CRC Press, 1999.
